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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
ARTICLE NO.
207, 73]95 Ž1997.
AY975262
On Neutral Functional]Differential Equations with
Proportional Delays
Arieh Iserles
Department of Applied Mathematics and Theoretical Physics, Uni¨ ersity of Cambridge,
Cambridge, England
and
Yunkang Liu*
Fitzwilliam College, Uni¨ ersity of Cambridge, Cambridge, England
Submitted by Thomas S. Angell
Received June 9, 1994
In this paper we develop a comprehensive theory on the well-posedness of the
initial-value problem for the neutral functional-differential equation
X
y Ž t . s ay Ž t . q
`
`
is1
is1
Ý bi y Ž qi t . q Ý ci yX Ž pi t . ,
t ) 0,
y Ž0. s y 0 ,
and the asymptotic behaviour of its solutions. We prove that the existence and
uniqueness of solutions depend mainly on the coefficients c i , i s 1, 2, . . . , and on
the smoothness of functions in the solution space. As far as the asymptotic
behaviour of analytic solutions is concerned, the c i have little effect. We prove that
if Re a ) 0 then the solution y Ž t . either grows exponentially or is polynomial. The
most interesting result is that if Re a F 0 and a / 0 then the asymptotic behaviour
of the solution depends mainly on the characteristic equation
`
aq
Ý bi qil s 0.
is1
These results can be generalized to systems of equations. Finally, we present some
examples to illustrate the change of asymptotic behaviour in response to the
variation of some parameters. The main idea used in this paper is to express the
solution in either Dirichlet or Dirichlet]Taylor series form. Q 1997 Academic Press
Current Address: Gonville and Caius College, University of Cambridge, Cambridge,
England.
73
0022-247Xr97 $25.00
Copyright Q 1997 by Academic Press
All rights of reproduction in any form reserved.
74
ISERLES AND LIU
1. INTRODUCTION
Initial-value problems for functional-differential equations of constant
time delays of the form
d
dt
ž
yŽ t. y
`
Ý c i y Ž t y pi .
is1
/
s ay Ž t . q
`
Ý bi y Ž t y qi . ,
t ) 0, Ž 1.1.
is1
y Ž t . s y0 Ž t . ,
x F 0,
Ž 1.2.
where a, bi , and c i are complex constants, pi , qi ) 0, i s 1, 2, . . . , satisfying sup i G 1 pi , sup iG 1 qi - `, and y 0 Ž t . is a given function, have been
studied extensively. It is well known that Ž1.1. ] Ž1.2., under some minor
restrictions, has a unique solution and the asymptotic behaviour of this
solution depends mainly on the corresponding characteristic equation
l 1y
ž
`
Ý c i exp Ž yl pi .
is1
/
saq
`
Ý bi exp Ž yl qi . .
is1
Details for this and for more general results can be found in Hale w11x. In
this paper we study the initial-value problem for neutral functional-differential equations of the form
yX Ž t . s ay Ž t . q
`
`
is1
is1
Ý byi Ž qi t . q Ý c i yX Ž pi t . ,
y Ž 0. s y 0 ,
t ) 0,
Ž 1.3.
Ž 1.4.
where a, bi , and c i are complex constants, pi , qi g Ž0, 1., i s 1, 2, . . . , and
y 0 is a given initial value. To guarantee convergence of the series in Ž1.3.,
we assume that Ý`is1 < bi <, Ý`is1 < c i < - `. A list of applications for this kind of
equation features in Iserles w12x. One remarkable difference between Ž1.1.
and Ž1.3. is that the latter has unbounded variable time delays.
The objective of this paper is to develop a fairly comprehensive theory
on the well-posedness of the initial-value problem Ž1.3. ] Ž1.4. and, most
importantly of all, the asymptotic behaviour of its solutions. We prove that
the existence and uniqueness of solutions of Ž1.3. ] Ž1.4. depend mainly on
the coefficients c i , i s 1, 2, . . . , and the smoothness of functions in the
solution space. However, the c i have little effect on the asymptotic
behaviour of analytic solutions of Ž1.3. ] Ž1.4.. We prove that if Re a ) 0
then the solution y Ž t . either grows exponentially, i.e., lim t ª` y Ž t . eya t / 0,
or is polynomial. Our most interesting result is that if Re a F 0 and a / 0
then the asymptotic behaviour of the solution depends mainly on the
characteristic equation
aq
`
Ý bi qil s 0.
is1
NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS
75
Our results can be generalized to systems of functional-differential equations with proportional time delays. Finally, we specialize our discussion by
paying closer attention to the generalized pantograph equation w12x
yX Ž t . s ay Ž t . q by Ž qt . q cyX Ž qt . ,
t ) 0,
y Ž 0 . s 1. Ž 1.5.
We present some examples that illustrate the change of asymptotic behaviour in response to the variation of some parameters.
2. EXISTENCE AND UNIQUENESS OF SOLUTIONS
Analogous to the existence and uniqueness theorem of Nussbaum w22x,
we have the following result Žsee also Iserles and Liu w14x and w17x..
THEOREM 1. If Ý`is1 pim < c i < - 1 for some m g Zq then in the function
space C mq 1 w0, `. the solution of Ž1.3. ] Ž1.4. exists and is unique if and only if
`
Ý pin c i / 1,
n s 0, 1, . . . , m y 1.
Ž 2.1.
is1
If Ž2.1. holds then the unique solution is analytic and can be written as
y Ž t . s y0 q y0
`
`
ny1
Ý Ł
ns1 ks0
ž
1y
Ý pik c i
is1
y1
/ ž
aq
`
Ý qik bi
is1
/
tn
n!
.
Ž 2.2.
Kuang and Feldstein w20x studied the scalar initial-value problem
yX Ž t . s ay Ž t . q
M
K
is1
is1
Ý bi y Ž qi t . q Ý c i yX Ž pi t . ,
t ) 0,
y Ž 0. s y 0 ,
Ž 2.3.
where a, bi , i s 1, . . . , M, and c i , i s 1, . . . , K, are real constants. By
transforming Ž2.3. into an integral equation, they proved that it has a
K <
< - 1. They
unique solution Žin the function space C 1 w0, `.. if Ý is1
c i py1
i
then posed the problem of establishing the same result in the case of
K <
K <
< G 1. Our result shows that the condition Ý is1
< - 1 can
Ý is1
c i py1
c i py1
i
i
K < <
be modified to Ý is1 c i - 1.
Next we discuss the necessity of the assumption Ý`is1 pim < c i < - 1 made in
Theorem 1. For simplicity, we only consider the case m s 0. First, we
introduce the following lemma, which is almost the same as a result of
Nussbaum w22x.
76
ISERLES AND LIU
LEMMA 2. Let p g Ž0, 1. be gi¨ en and consider the homogeneous functional equation
y Ž t . s cy Ž pt . ,
t G 0,
Ž 2.4.
in the function space C w0, `..
1.
2.
3.
tions of
If < c < F 1 but c / 1 then Ž2.4. has only the tri¨ ial solution y Ž t . ' 0.
If c s 1 then all solutions of Ž2.4. are constant.
If < c < ) 1 then there is a one-to-one correspondence between soluŽ2.4. and functions in the space CU s f Ž t . g C w p, 1x: f Ž1. s cf Ž p .4 .
Proof. The first two parts of Lemma 2 are easy to prove. Consider part
3. For every solution y Ž t . g C w0, `. of Ž2.4., there corresponds a function
f Ž t . s y Ž t ., t g w p, 1x, which belongs to CU . On the other hand, for every
function f Ž t . g CU , it is easy to see that the function y Ž t . given by the
following recurrence relation:
y Ž 0 . s 0,
y Ž t . s c k f Ž pyk t . ,
p kq 1 F t F p k ,
k s 0, " 1, " 2, . . . ,
is a continuous solution of Ž2.4..
EXAMPLE 1. Consider solutions of the initial-value problem
yX Ž t . s cyX Ž pt . ,
t ) 0,
y Ž 0. s y 0 ,
Ž 2.5.
in the function space C 1 w0, `.. It follows from Lemma 2 that
1. The only solution of Ž2.5. is y Ž t . ' y 0 if < c < F 1 but c / 1.
2. All the solutions of Ž2.5. can be written as y Ž t . s y 0 q y 1 t if
c s 1, where y 1 is an arbitrary constant.
3. There is a one-to-one correspondence between solutions of Ž2.5.
and functions in the space CU if < c < ) 1.
This example shows that the initial-value problem Ž2.5. loses uniqueness
of solutions if either c s 1 or < c < ) 1.
EXAMPLE 2. Consider the homogeneous problem
yX Ž t . s
`
Ý c i yX Ž pi t . ,
t ) 0,
y Ž 0 . s 0.
Ž 2.6.
is1
Suppose that Ý`is1 c i ) 1. By the intermediate-value theorem there exists a
constant l ) 1 such that Ý`is1 pilc i s 1. Hence, the homogenous problem
Ž2.6. has the nontrivial solution y Ž t . s t lq1.
NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS
77
EXAMPLE 3. Consider solutions of the initial-value problem
yX Ž t . s c1 yX Ž p 1 t . q c 2 yX Ž p 2 t . y c1 c 2 yX Ž p 1 p 2 t . ,
t ) 0,
y Ž 0. s y 0 ,
Ž 2.7.
in the function space C 1 w0, `.. Suppose that < c1 < F 1 and < c 2 < F 1 but
c1 , c 2 / 1. Let y Ž t . g C 1w0, `. be a solution of Ž2.7.. Then x Ž t . s yX Ž t . y
c1 yX Ž p1 t . satisfies the functional equation
x Ž t . s c2 x Ž p2 t . ,
t 2 G 0.
Lemma 2 implies that x Ž t . ' 0. Therefore,
yX Ž t . s c1 yX Ž p 1 t . ,
t G 0,
y Ž 0. s y 0 .
It follows from Lemma 2 that yX Ž t . ' 0. Hence, the only C 1w0, `. solution
of Ž2.7. is y Ž t . ' y 0 . This example shows that the condition < c1 < q < c 2 < q
< c1 c 2 < - 1 is unnecessary for the uniqueness of solution of Ž2.7. in the
function space C 1 w0, `..
In the rest of this paper, we assume that a / 0, that Ž2.1. holds, and that
lqs sup iG 1 max pi , qi 4 - 1. We shall only consider the analytic solution
of Ž1.3. ] Ž1.4..
3. DIRICHLET AND DIRICHLET]TAYLOR SERIES SOLUTIONS
The Taylor series solution Ž2.2. is of little use in the study of asymptotic
behaviour. On the other hand, it seems that the solution cannot be found
through Laplace transform as was done in the case of Ž1.1. ] Ž1.2.. Fortunately, it can be expressed in either Dirichlet or Dirichlet]Taylor series
form. To begin with, we introduce some notation. For sequences x i 4`is1 ,
yi 4`is1 of complex numbers and ji 4`is1 of integers, we define
x s Ž x 1 , x 2 , . . . . g C` ,
y s Ž y 1 , y 2 , . . . . g C` ,
j s Ž j1 , j2 , . . . . g Z` ,
and the standard multi-index operations
<x < s
`
Ý < xi <,
<j < s
is1
`
Ý < ji < ,
is1
Ž x, y . s
xjs
`
Ł x ij ,
i
is1
`
Ý x i yi ,
is1
xy s Ž x 1 y 1 , x 2 y 2 , . . . . ,
x k s Ž x 1k , x 2k , . . . . ,
k g Zq.
78
ISERLES AND LIU
First, we seek Dirichlet series solution of the form
y Ž t . s a y0
`
Ý
Ý
ns0 <j <q <k <sn
d j, k exp Ž atq j p k . ,
Ž 3.1.
where a and d j, k , j, k g ŽZq. ` , are coefficients to be determined. It is
understood that pi s 1 Ž qi s 1. if c i s 0 Ž bi s 0. for some i g N. Let
d 0, 0 s 1, where 0 s Ž0, 0, . . . ., and d j, k s 0 if ji - 0 or k i - 0 for some
i g N. It is easy to see that
`
yX Ž t . s a a
Ý
Ý
ns0 <j <q <k <sn
q j p k d j, k exp Ž atq j p k . ,
and for i g N that
y Ž qi t . s a y 0
`
Ý
Ý
ns0 <j <q <k <sn
yX Ž pi t . s a a y 0
d jye i , k exp Ž atq j p k . ,
`
Ý
Ý
ns0 <j <q <k <sn
q j p kye i d j, kye i exp Ž atq j p k . ,
where e i s Ž d 1, i , d 2, i , . . . . with dn, i denoting the Kronecker symbol. By
substituting Ž3.1. into Ž1.3. and comparing the coefficients of expŽ atq j p k .
on both sides, we obtain the recurrence relation
`
Ž q j p k y 1 . d j, k s Ý
is1
ž
bi
a
d jye i , k q q j p kye i c i d j, kye i ,
/
`
j, k g Ž Zq . ,
Ž 3.2.
which leads to the following result.
THEOREM 3. If <b < - < a < then the solution of Ž1.3. ] Ž1.4. has a Dirichlet
series expansion of the form Ž3.1., where d j, k 4 are determined by Ž3.2. and
as
`
Ł
1 q Ž b, q n . ra
ns0
1 y Ž c, p n .
.
Ž 3.3.
Proof. It is not difficult to see from Ž3.2. that
< d j, k < F r
max
<l <q <m <- <j <q <k <y1
< d l, m < ,
`
j, k g Ž Zq . ,
for r s Ž<b <r< a < q <c <.rŽ1 y lq ., which implies that
< d j, k < F r < j<q< k < ,
`
j, k g Ž Zq . .
NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS
79
Therefore, the generating function
g Ž x, y . s
`
Ý
Ý
ns0 <j <q <k <sn
d j, k x j y k
is analytic for all x, y g C` satisfying <x < q <y < - r . Multiplying Ž3.2. by x j y k
and summing up for j, k g ŽZq. ` yields
g Ž qx, py . y g Ž x, y . s
Ž b, x .
a
g Ž x, y . q Ž c, y . g Ž qx, py . ,
which implies that
g Ž x, y . s
1 y Ž c, y .
1 q Ž b, x . ra
g Ž qx, py . .
Due to the fact that lim x, y ª 0 g Žx, y. s g Ž0, 0. s 1, we deduce that
g Ž x, y . s
`
Ł
ns0
1 y Ž c, p n y .
1 q Ž b, q n x . ra
,
Ž 3.4.
where the convergence of the product follows from our assumption that
lq- 1. ŽNote that formula Ž3.4. is, in fact, a generalization of the q-binomial theorem w10x. A further generalization is Ž5.9., which features in
Section 5.. Hence, g Žx, y. can be extended analytically to x, y g C` :
<b < <x < - < a <4 . Since < a < ) <b <, we deduce that the series Ý`ns 0 Ý < j<q< k <sn d j, k
converges absolutely. Therefore, the Dirichlet series Ž3.1. converges absolutely for all t g R. Our assumption that Ž2.1. holds implies that g Ž1, 1. / 0,
where 1 s Ž1, 1, . . . .. It is easy to verify that the Dirichlet series Ž3.1.
satisfies Ž1.3. ] Ž1.4. if we let a s 1rg Ž1, 1..
The Dirichlet series may fail to converge if <b < G < a <. However, we can
express a high-order derivative of the solution of Ž1.3. ] Ž1.4. into a Dirichlet series, and find the solution through the Taylor expansion formula with
integral remainder. Let m g N be such that <Žb, q m .< - < a <, and let ymŽ t . be
the mth derivative of the solution y Ž t . of Ž1.3. ] Ž1.4.. Therefore,
my1
yŽ t. s
Ý
yn Ž 0 .
ns0
n!
tn q
1
t
Ž m y 1. !
H0 Ž t y t .
my1
ym Ž t . dt ,
Ž 3.5.
where, by Theorem 1,
ny1
yn Ž 0 . s y 0
Ł
ks0
ž
1y
y1
`
Ý
is1
pik c i
/ ž
aq
`
Ý qik bi
is1
/
,
n s 1, 2, . . . . Ž 3.6.
80
ISERLES AND LIU
Differentiating both sides of Ž1.3. m times yields
X
ym
Ž t . s aym Ž t . q
`
Ý
bi qim ym Ž qi t . q
is1
`
Ý c i pim ymX Ž pi t . ,
t ) 0. Ž 3.7.
is1
According to Theorem 3, we have
ym Ž t . s a m ym Ž 0 .
`
Ý
Ý
ns0 <j <q <k <sn
d j, k , m exp Ž atq j p k . ,
where d 0, 0, m s 1, d j, k, m s 0 if ji - 0 or k i - 0 for some i g N, d j, k, m 4
satisfies the recurrence relation
`
Ž q j p k y 1 . d j, k , m s Ý
is1
ž
bi qim
a
d jye i , k , m q q j p kye i c i pim d j, kye i , m ,
/
`
j, k g Ž Zq . ,
and a m satisfies a m ymŽ0. s a ma y 0 . Noting that d j, k, m s Žq j p k . m d j, k and
1
t
H Žtyt.
Ž m y 1. ! 0
my 1
e at dt s aym e at y
ž
my1
Ý
ls0
al
l!
/
tl ,
we have the following result.
THEOREM 4. If m [ min n g Zq: <Žb, q n .< - < a <4 G 1, then the solution
of Ž1.3. ] Ž1.4. has the Dirichlet]Taylor series expansion
my1
yŽ t. s
y Ž n. Ž 0 .
Ý
ns0
q a y0
n!
tn
`
Ý
Ý
ns0 <j <q <k <sn
ž
my1
j
d j, k exp Ž atq p
k
.y Ý
ls0
Ž aq j p k .
l!
l
/
tl ,
where y Ž n. Ž0., a and d j, k 4 are gi¨ en by Ž3.6., Ž3.3., and Ž3.2., respecti¨ ely.
4. ASYMPTOTIC BEHAVIOUR
In this section we demonstrate how to use Dirichlet and Dirichlet]Taylor
series solutions to analyse the asymptotic behaviour of the neutral equation Ž1.3..
NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS
81
THEOREM 5. Let y Ž t . be the unique solution of Ž1.3. ] Ž1.4. and m [
min n g Zq: <Žb, q n .< - < a <4 . Then
1. lim t ª` y Ž t . eya t s a y 0 if Re a ) 0, where a is gi¨ en by Ž3.3.;
2. y Ž t . s O Ž t m . as t ª ` if Re a s 0. Moreo¨ er, y Ž m. Ž t . is almost
periodic; and
3. y Ž t . s oŽ t m . as t ª ` if Re a - 0.
Proof. We only give proof of the case of Re a ) 0, since the other two
cases can be proved similarly. If m s 0 then we deduce from Ž3.1. that
y Ž t . exp Ž yat . y a y 0 F exp Ž yRe a Ž 1 y lq . t .
`
Ý
Ý
ns1 <j <q <k <sn
< d j, k < ,
which implies that lim t ª` y Ž t . eya t s a y 0 . If m G 1 then we deduce from
Ž3.7. that lim t ª` ymŽ t . eya t s a m ymŽ0. s a ma y 0 . Using the Taylor expansion Ž3.5., we deduce that lim t ª` y Ž t . eya t s a y 0 .
Part 1 of the last theorem shows that the solution y Ž t . either grows
exponentially, i.e., lim t ª` y Ž t . eya t / 0, or is polynomial.
THEOREM 6.
Let y Ž t . be the unique solution of Ž1.3. ] Ž1.4. and let
¡sup Re l: a q Ý b q
½
[~
¢y`,
`
m0
1.
w0, `..
2.
i
l
i
If Re a s 0 then y Ž t . s O Ž t
m.
5
s0 ,
is1
b / 0,
b s 0.
as t ª ` for all m g Ž m 0 , `. l
If Re a - 0 then y Ž t . s oŽ t m . as t ª ` for all m ) m 0 .
To prove the theorem, we introduce three lemmas.
LEMMA 7.
Suppose that y Ž t . g C w0, `. satisfies the functional equation
ay Ž t . q
`
Ý bi y Ž qi t . s h Ž t . ,
t G 0,
is1
where a, bi , and qi , i s 1, 2, . . . , are as before, and hŽ t . g C w0, `. satisfies
hŽ t . s O Ž t m1 . as t ª ` for some constant m 1. Then y Ž t . s O Ž t m . as t ª `
for any m g Ž m 0 , `. l w m 1 , `., where m 0 is the same as in Theorem 6.
Proof. For any m g Ž m 0 , `. l w m 1 , `. we define x Ž t . s eym t y Ž e t ., y`
- t - `. The function x Ž t . satisfies the difference equation
ax Ž t . q
`
Ý bi qim x Ž t q log qi . s H Ž t . ,
is1
0 F t - `,
82
ISERLES AND LIU
where H Ž t . [ ey m t hŽ e t . s O Ž1. as t ª `. Since sup Re l: a q
Ý`is1 bi qim q l s 04 s m 0 y m - 0, it follows from Theorem 4.1 of Hale w11,
p. 287x that x Ž t . s O Ž1. as t ª `. Therefore, y Ž t . s O Ž t m . as t ª ` for
any m g Ž m 0 , `. l w m 1 , `..
LEMMA 8. If b , l g Ž0, 1. then Ý`ns 0 b n expŽyt l n . s O Ž t m . but not
oŽ t m . as t ª ` for m s ylog brlog l - 0.
Proof. If m F y1 then
ty m
`
Ý
b n exp Ž yt l n . s
ns0
l1q m
1yl
1q m
F
-
l
1yl
l1q m
`
ym y1
exp Ž yt l n . Ž t l n y t l nq1 .
Ý Ž t lnq1 .
ns0
`
tln
ÝH
nq1
ns0 t l
`
ym y1
Hx
1yl 0
xym y1 exp Ž yx . dx
exp Ž yx . dx - `,
and, for t G l,
ty m
`
Ý
b n exp Ž yt l n . s
ns0
lym
1yl
ym
G
l
1yl
`
ym y1
exp Ž yt l n . Ž t l ny1 y t l n .
Ý Ž t lny1 .
ns0
`
t l ny1 ym y1
ÝH
n
ns0 t l
x
exp Ž yx . dx
lym
xy m y1 exp Ž yx . dx ) 0.
1yl
Hence, Ý`ns 0 b n expŽyt l n . s O Ž t m . but not oŽ t m . as t ª `. The case of
m g Žy1, 0. can be proved similarly.
)
Alternatively, we can prove the preceding lemma in the following way.
The function
`
1
xŽ t. [ Ý
b n exp Ž yt l n .
ns0 Ž l ; l . n
satisfies the functional-differential equation
xX Ž t . s b x Ž l t . y x Ž t . ,
t ) 0.
ŽIn what follows we use standard notation from the theory of basic
hypergeometric functions w10x, e.g.,
Ž a; q . n s
½
1,
Ž 1 y a. Ž 1 y aq . ??? Ž 1 y aq ny 1 . ,
n s 0,
n s 1, 2, . . . ,
called the shifted factorial or q-factorial or Gauss]Heine symbol, and
Ž a; q .` s Ł`ks0 Ž1 y aq k ...
NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS
83
According to Theorem 3Ži. and Žii. of Kato and McLeod w19x, x Ž t . s
O Ž t m . but not oŽ t m . as t ª `. Hence, Lemma 8 holds since
Ž l ; l. ` x Ž t . -
`
b n exp Ž yt l n . - x Ž t . .
Ý
ns0
In the remainder of this section we let ly[ inf iG 1 max pi , qi 4 ) 0.
LEMMA 9. If Re a - 0 and <b < - < a < then y Ž t . s O Ž t m . as t ª ` for
m ) Žlog < a < y log <b <.rlog ly.
Proof. It is easy to see from Ž3.2. that for any e g Ž0, 1 y <b <r< a <. there
exists a constant M ) 0 such that
Ý
<j <q <k <sn
< d j, k < F M
ž
<b < q e
< a<
n
/
,
n G 0.
According to Theorem 3, we have
y Ž t . F < a y0 < M
`
Ý
ns0
ž
<b < q e
< a<
n
/
n
exp Ž Re at ly
..
It follows from Lemma 8 that y Ž t . s O Ž t m . as t ª ` for m s Žlog < a < y
logŽ<b < q e ..rlog ly. The arbitrariness of e implies the desired result.
Proof of Theorem 6. Denote ynŽ t . s y Ž n. Ž t ., n g N. An important observation is that ynŽ t . satisfies the functional-differential equation
ynX Ž t . s ayn Ž t . q
`
`
is1
is1
Ý bi qin yn Ž qi t . q Ý c i pin ynX Ž pi t . ,
t ) 0, Ž 4.1.
and the functional equation
ayn Ž t . q
`
Ý bi qin yn Ž qi t . s h n Ž t . ,
t G 0,
Ž 4.2.
is1
where
h n Ž t . s ynq1 Ž t . y
`
Ý c i pin ynq1Ž pi t . .
is1
Another useful observation is that ynq 1Ž t . s O Ž t m . as t ª ` implies
h nŽ t . s O Ž t m . as t ª ` for any fixed m and n. First, let us consider the
case Re a s 0. Let m g N be such that <Žb, q m .< - < a <. Applying Theorem 5
to Ž4.1. with n s m yields ymŽ t . s O Ž1. as t ª `. Applying Lemma 7
84
ISERLES AND LIU
repeatedly to Ž4.2. from n s m y 1 to n s 0 yields y Ž t . s O Ž t m . as t ª `
for any m g Ž m 0 , `. l w0, `.. Next, we consider the case Re a - 0. Let
m g N be such that
log < a < y log Ž b, q m .
log ly
- m0 .
U
Applying Lemma 9 to Ž4.1. with n s m yields ymŽ t . s O Ž t m . as t ª ` for
any
mU g
ž
log < a < y log Ž b, q m .
log ly
/
, m0 .
Applying Lemma 7 repeatedly to Ž4.2. from n s m y 1 to n s 0 yields
y Ž t . s O Ž t Ž m q m 0 .r2 . as t ª ` for any m ) m 0 . Hence, y Ž t . s oŽ t m . as
t ª ` for any m ) m 0 . This completes the proof of Theorem 6.
5. ON A SYSTEM OF EQUATIONS
In this section we generalize our results on the scalar initial-value
problem Ž1.3. ] Ž1.4. to the vector problem
y X Ž t . s Ay Ž t . q
`
Ý
is1
y Ž 0 . s y0 ,
Bi y Ž qi t . q
`
Ý Ci y X Ž pi t . ,
t ) 0,
Ž 5.1.
is1
Ž 5.2.
where A, Bi , and Ci are d = d complex matrices, pi , qi g Ž0, 1., i s
1, 2, . . . , and y0 is a column vector in C d. To guarantee convergence of the
series in Ž5.1., we assume that Ý`is1 5 Bi 5 - ` and Ý`is1 5 Ci 5 - `, where 5 ? 5
denotes the matrix norm induced by a vector norm, likewise denoted by
5 ? 5, which is arbitrary. Note that this assumption is independent of the
norm, since all norms in a finite-dimensional space are equivalent. In the
sequel s Ž?. denotes the spectrum, r Ž?. the spectral radius, a Ž?. the
maximal real part of the eigenvalues of the matrix Žthe spectral abscissa.,
and k Ž A. the geometric multiplicity of the eigenvalue with maximal real
part. If there is more than one eigenvalue that attains the maximal real
part, then k Ž A. denotes the maximal geometric multiplicity of all such
eigenvalues.
NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS
85
Analogously to Theorem 1 we have the following result.
THEOREM 10. If Ý`is1 pim 5 Ci 5 - 1 for some m g Zq, then in the function space C mq 1 w0, `. the solution of Ž5.1. ] Ž5.2. exists and is unique if and
only if
`
Iy
Ý pin Ci , n s 0, 1, . . . , m y 1, are nonsingular.
Ž 5.3.
is1
If Ž5.3. holds then the unique solution is analytic and can be written as
ž
`
yŽ t . s I q
ny1
Ý Ł
ns1 ks0
Iy
ž
y1
`
Ý
pik Ci
is1
/ ž
Aq
`
Ý qik Bi
is1
tn
/ /
n!
t G 0,
y0 ,
Ž 5.4.
where the multiplication of matrices in the product is carried out from left to
right.
In the rest of this section, we assume that A is nonsingular, Ž5.3. holds,
lqs sup iG 1 max pi , qi 4 - 1, and
s Ž A . l s Žq j p kA . s B
Ž 5.5.
holds for all Žj, k. g ŽZq. ` = ŽZq. ` _Ž0, 0.. Again we seek a Dirichlet or
Dirichlet]Taylor series solution.
Analogously to Theorem 3 we have the following result.
THEOREM 11.
series solution
If Ý`is1 5 Ay1 Bi 5 - 1 then Ž5.1. ] Ž5.2. has the Dirichlet
yŽ t . s
`
Ý
D j, k exp Ž Atq j p k . V y0 ,
Ý
ns0 <j <q <k <sn
Ž 5.6.
where D 0, 0 s I, D j, k s 0 if ji - 0 or k i - 0 for some i g N, D j, k 4 , Žj, k. g
ŽZq. ` = ŽZq. ` _Ž0, 0., are determined by the recurrence relation
q j p k D j, k A y ADj, k
s
`
Ý Ž Bi Djye , k q q j p kye Ci Dj, kye A . ,
i
i
is1
i
`
j, k g Ž Zq . , Ž 5.7.
86
ISERLES AND LIU
and
ym
my1
V s lim A
Ł
mª`
ns0
ž
Iy
y1
`
Ý
Ci pin
is1
/ ž
Aq
`
Ý Bi qin
is1
/
.
Ž 5.8.
Proof. By substituting Ž5.6. into Ž5.1. and comparing the coefficients of
expŽ Atq j p k . on both sides, we obtain the recurrence relation Ž5.7.. Clearly,
Ž5.7. is obeyed in the case j, k s 0. It follows from a classical theorem w9x
that Ž5.7. is solvable for D j, k if Ž5.5. holds. Introducing the generating
function
G Ž x, y . s
`
Ý
Ý
ns0 <j <q <k <sn
D j, k x j y k ,
we obtain from Ž5.7. the relation
ž
`
Aq
Ý Bi x i
is1
/
G Ž x, y . s I y
ž
`
Ý C i yi
is1
/
G Ž qx, py . A,
which implies that
det Ž G Ž x, y . . s
`
Ł
ns0
det Ž I y Ý`is1Ci pin yi .
det Ž I q Ý`is1TAy1 Bi qin x i .
/ 0.
Ž 5.9.
Hence, GŽ1, 1. is nonsingular. By letting V s GŽ1, 1.y1 , we see that the
Dirichlet series Ž5.6. satisfies Ž5.1. ] Ž5.2..
Analogously to Theorem 4 we have the following result.
THEOREM 12. If m [ min n g Zq: Ý`is1 qin 5 Ay1 Bi 5 - 14 G 1, then
Ž5.1. ] Ž5.2. has the Dirichlet]Taylor series solution
my1
yŽ t . s
y Ž n. Ž 0 .
Ý
ns0
q
n!
tn
`
Ý
Ý
ns0 <j <q <k <sn
ž
D j, k exp Ž Atq j p k . y
my1
Ý
ls0
Ž Aq j p k .
l!
l
/
t l V y0 ,
where y Ž n. Ž0. can be obtained from Ž5.4., D j, k are determined by Ž5.7., and V
is gi¨ en by Ž5.8..
NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS
87
Noting that
5 e A t 5 F M Ž t q 1 . k Ž A .y1 e a Ž A.t ,
t G 0,
for some constant M ) 0, we have analogously to Theorems 5 and 6 the
following results.
THEOREM 13.
1.
2.
3.
Let yŽ t . be the unique solution of Ž5.1. ] Ž5.2.. Then
5yŽ t .5 s O Ž t k Ž A.y1 e a Ž A.t . as t ª ` if a Ž A. ) 0;
5yŽ t .5 s O Ž t mq k Ž A.y1 . as t ª ` if a Ž A. s 0; and
5yŽ t .5 s oŽ t m . as t ª ` if a Ž A. - 0,
where m [ min n g Zq: Ý`is1 qin 5 Ay1 Bi 5 - 14 .
THEOREM 14.
if
Suppose that ly[ inf iG 1 max pi , qi 4 ) 0. Let m 0 s y`
det A q
ž
`
Ý Bi qil
is1
/
s0
has no solution; otherwise, let
½
m 0 [ sup Re l : det A q
ž
`
Ý Bi qil
is1
/ 5
s0 .
Then the solution yŽ t . of Ž5.1. ] Ž5.2. satisfies
1. 5yŽ t .5 s O Ž t mq k Ž A.y1 . as t ª ` for any m g Ž m 0 , `. l w0, `. if
a Ž A. s 0; and
2. 5yŽ t .5 s oŽ t m . as t ª ` for any m ) m 0 if a Ž A. - 0.
Under further restrictions, we have the following result.
THEOREM 15. If all eigen¨ alues of A ha¨ e a positi¨ e real part, and A, Bi ,
and Ci commute with each other for all i g N, then the solution yŽ t . of
Ž5.1. ] Ž5.2. satisfies
lim eyA t y Ž t . s V y0 ,
tª`
where V is gi¨ en by Ž5.8..
Proof. It is easy to prove by induction from Ž5.7. that, in the present
situation, A and D j, k 4 , Žj, k. g ŽZq. ` = ŽZq. ` , commute with each other.
88
ISERLES AND LIU
We can then prove this theorem directly from Theorem 11 or 12, depending on whether the condition Ý`is1 5 Ay1 Bi 5 - 1 holds or not.
6. ON THE GENERALIZED PANTOGRAPH EQUATION
In this section we discuss the generalized pantograph equation Ž1.5.. To
guarantee the existence and uniqueness of the solution, we assume that
c / qyk , k s 0, 1, . . . . Suppose that a / 0 and < b < - < a <. Then Ž1.5. has the
Dirichlet series solution
yŽ t. s
Ž ybra; q . `
Ž c; q . `
`
Ý
ns0
b
Ž yacrb; q . n
y
a
Ž q; q . n
ž /
n
exp Ž atq n .
if b / 0, or
yŽ t. s
n
1
`
Ž c; q . `
ns0
Ý
Ž y1. c nq nŽ ny1.r2
exp Ž atq n .
Ž q; q . n
if b s 0. Suppose that < b < G < a < ) 0. Then Ž1.5. has the Dirichlet]Taylor
series solution
my1
yŽ t. s
y Ž n. Ž 0 .
Ý
ns0
=
`
Ý
ns0
n!
tn q
Ž ybra; q . `
Ž c; q . `
b
Ž yacrb; q . n
y
a
Ž q; q. n
n
ž /ž
my1
exp Ž atq n . y
Ý
ls0
Ž aq n .
l!
l
/
tl ,
where m [ wlogŽ< a <r< b <.rlog q x q 1.
In the case where Re a - 0 and b / 0, the result in Section 4 can be
improved.
THEOREM 16. Suppose that Re a - 0 and b / 0. Let y Ž t . be the solution
of Ž1.5., m s logŽ< a <r< b <.rlog q, u s 1rŽ2p .argŽyq may1 b ., and z Ž t . s
tym y Ž t .. Then
1. y Ž t . s O Ž t m . as t ª `.
2. If u is rational, u s lrk, say, then the v-limit set of z Ž t . coincides
with a closed continuous cur¨ e that can be represented parametrically by
ŽRe zU Ž t ., Im zU Ž t .. for t g w1, qyk x in the complex plane, where zU Ž t . g
NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS
FIG. 1.
89
X
. Ž .
Ž
. Ž .
y Ž t . s expŽ 11
20 p i y t y y tr4 , y 0 s 1.
C w1, qyk x satisfies
zU Ž t . s e 2 pu i zU Ž qt . ,
t g qy1 , qyk .
3. If u is irrational then the v-limit set of z Ž t . is the annulus V s z:
z F < z < F z 4 , where z lim inf t ª` < z Ž t .<, z s lim sup t ª` < z Ž t .<.
This theorem is proved in Liu w16x for a slightly more general case. The
Dirichlet and Dirichlet]Taylor series solutions make it possible to calculate the v-limit set of z Ž t . Žthe numerical methods presented in Liu w17x
can be used in more general cases.. In Figs. 1]6 we display the solutions
for t c 1 in the ‘‘phase plane’’ ŽRe y, Im y ..
Finally, we discuss the change in asymptotic behaviour in response to
the variation of some parameters. Denote by y Ž t; q . the solution of Ž1.5.. It
90
ISERLES AND LIU
FIG. 2.
X
. Ž .
Ž
. Ž .
y Ž t . s expŽ 11
20 p i y t y y tr10 , y 0 s 1.
is easy to see that
y Ž t ; 1 . s exp
ž
aqb
1yc
t
/
and
y Ž t ; 0. s y
c q bra
1yc
q
1 q bra
1yc
e at .
Consider the case of Re a ) 0. Obviously, the asymptotic behaviour of
y Ž t; q . is different from y Ž t; 1. no matter how close q is to 1 except when
NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS
91
X
. Ž
. Ž .
y Ž t . s expŽ 116 p i . y Ž t . q expŽ 16
11 p i y tr20 , y 0 s 1.
FIG. 3.
b s yac. Noting that
lim Ž 1 y q . log Ž a; q . ` s
qª1y
1
H0 logŽ 1 y ax .
dx
x
,
we obtain from Theorem 5 that
lim
qª1y
ž lim y Ž t ; q . expŽ yat . /
1y q
tª`
s exp
1
log 1 q
žH ž ž
0
bx
a
/
y log Ž 1 y cx .
dx
/ /
x
.
92
ISERLES AND LIU
FIG. 4.
X
. Ž y7 t ., y Ž0. s 1.
y Ž t . s expŽ 116 p i . y Ž t . q expŽ 16
11 p i y 10
However, as q ª 0 q , the asymptotic behaviour of y Ž t; q . is quite similar
to y Ž t; 0.. In fact, we have
lim lim y Ž t ; q . eya t s lim y Ž t ; 0 . eyat s
qª0q tª`
tª`
1 q bra
1yc
.
Consider the case of Re a - 0. If we treat < bra< as a parameter, it is
clear from Theorem 16 that there exists a Hopf-like bifurcation as < bra<
passes through 1. Another interesting phenomenon happens when < b < s < a <
and a ª 0 q . It seems that the v-limit set of y Ž t; q . tends to a set Žsee
Figures 3 and 4. which is obivously distinct from yŽ c q bra.rŽ1 y c ., the
limiting value of y Ž t; 0..
NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS
FIG. 5.
93
X
. Ž .
Ž 12 . Ž
. Ž .
y Ž t . s expŽ 40
41 p i y t q exp 41 p i y tr2 , y 0 s 1.
Remark. The case a s 0 has been studied by Morris, Feldstein, and
Bowen w21x, Kuang and Feldstein w20x, Iserles w12x, and Liu w16x. In the first
two papers the authors proved the unboundedness of the nonpolynomial
solution via the Phragmen]Lindelof
´
¨ principle, and by the Ahlfors theorem
in the third paper. In the fourth paper, the author proved by using the
same technique as that used in Derfel w5x that for every nonpolynomial
solution y Ž t . of Ž1.3. there exists a sequence t n4`ns1 , which tends to ` as
n ª `, such that < y Ž t .< G M ŽexpŽ b Žlog t . 2 .. at t s t n , n s 1, 2, . . . , for
some positive constants M and b .
ACKNOWLEDGMENTS
The second author wishes to thank Professor Chunlai Zhao ŽPeking University. for a
helpful discussion on the proof of Lemma 8, and Cambridge Overseas Trust, CVCP,
Fitzwilliam College, and C. T. Taylor Fund for their financial support.
94
ISERLES AND LIU
FIG. 6.
X
. Ž .
Ž 20 . Ž
. Ž .
y Ž t . s expŽ 22
41 p i y t q exp 41 p i y tr2 , y 0 s 1.
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